| 58 | |
| 59 | 9.3 Thurstone’s Law of Comparative Judgment |
| 60 | |
| 61 | |
| 62 | Thurstone’s Law of Comparative Judgment (TLCJ) measures a person’s preference for one item i over any another item j based on a single stimulus measuring the discriminal dispersion between the two on a psychological continuum. This is conducted by using paired comparisons where every item i in a pair (i,j) in a set will be compared to every other item in a set producing a total of n(n-1)/2 comparisons (Thurstone, 1927). |
| 63 | |
| 64 | Torgerson provides an example of this process using three matrices: matrix F (frequency), matrix P (probability) and matrix X (unit normal derivative). The first matrix F, is a frequency count of item selected in any given pair by every individual in the group. The table presented next shows a frequency count of a user group where N = 5. The items along the 1st row, A, B, C, and D are preferred over the corresponding items in the 1st column of an M x M matrix, where zeros are placed along the diagonal. Given the number of users, N = 5, each corresponding pair should equal N, i.e. (i,j) + (j,i) = 5: |
| 65 | |
| 66 | A B C D |
| 67 | A ---- 1 2 1 |
| 68 | B 4 ---- 3 4 |
| 69 | C 3 2 ---- 5 |
| 70 | D 4 1 1 ---- |
| 71 | Table x.1 Matrix F Frequency Count of User Input |
| 72 | |
| 73 | These frequencies are changed to probabilities as such for matrix P where each each (i,j) + (j,i) = 100%: |
| 74 | |
| 75 | A B C D |
| 76 | A ---- .20 .40 .20 |
| 77 | B .80 ---- .60 .80 |
| 78 | C .60 .40 ---- 1.00 |
| 79 | D .80 .20 .0 ---- |
| 80 | 2.2 .80 1.0 2.0 |
| 81 | Table x.2 Matrix P Frequencies Converted to Probabilities |
| 82 | |
| 83 | The final matrix, X, is where these percentages are replaced by their unit normal deviates to cumulative proportions. |
| 84 | |
| 85 | A B C D |
| 86 | A ---- -.85 -.26 -.85 |
| 87 | B .84 ---- .25 .84 |
| 88 | C .25 -.26 ---- 0 |
| 89 | D .84 -.85 0 ---- |
| 90 | 1.93 -1.96 -.01 -.01 |
| 91 | Table x.3 Matrix X Cumulative Normal Distribution Function |
| 92 | |
| 93 | Values greater than 50% will be positive in this transformation of matrix P to X and values less than 50% will be negative. Any values of 100% or 0% in matrix P are given a value of 0 in matrix X because the x values corresponding from the unit normal deviate table are unboundedly large (Torgerson, 1958). |
| 94 | |
| 95 | Thurstone’s scale is an extension based off of Thorndike’s work which provides added insight into ranked lists of items. Given A is preferred over B 75% of the time and B is preferred over C 85% of the time, “how much greater than the distance AB is the distance BC? Thorndike solved this problem by assuming that the difference in distances is proportional to the difference in the unit normal deviates corresponding to the two proportions” (Torgerson, 1958, p. 155). |
| 96 | |